Quantum superposition

Before studying quantum mechanics, the idea that something can be two things at the same time looks quite weird. Let's try to convince ourselves that it's not.

Let's imagine we have a box full of balls of different colors.
We cannot see inside the box, but we can pull out a ball from the case and see its color.
Not being able to see all the balls inside, we do not know which color the ball we pull out it's going to be until we see it, but we know for sure that if we see a red ball, that must have been a red ball even when it was inside the box.
If we want to make a prediction on which color will be the next ball, we can define the probability like the ratio between the total number of balls extracted and the number of extracted balls of that color:
\[P_{\text{red ball}}=\frac{\text{Number of red balls extracted}}{\text{Number of balls extracted}}.\]
This is how classical probability works.

Let's now imagine that we have an LCD ball, just one, inside the box. When we put the ball in the box, where we can't see it, a device switches on and starts to randomly show patterns of any allowed color, even together!
When we pull the ball out, the device detects we grabbed it, and not only it stops mixing colors, but it also makes all the balls of one color among those allowed.
This is how quantum probability works.

From our point of view, there is no difference between the two systems, because when we watch a ball, we see a color, and we can measure the probability of each color to be seen. Nevertheless, there is a deeper difference.

While we could in principle break the classical box and count every single ball to know the probability, we can not break our LCD ball, because the probability of seeing a color is not given by our ignorance of the details of our system (aka the number of the balls of each color) but it is an a priori probability, which is intrinsic of the ball.

Why would be a cat both "dead and alive" then?

Because its life or death depends on a system whose laws are described by quantum mechanics.
The system, just like the LCD ball, can be in a superposition of more states at the same time, hence both switched on and off, black and white, dead and alive.

Note that it is just a paradox that the cat is either dead or alive, and the reason is that quantum mechanics does not work on systems bigger than a bunch of atoms.